For a Banach function lattice \(X\) , a function \(f \in X\)  is said to be order-continuous \(\Vert f_n\Vert _X \downarrow 0\)  for every sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  of measurable functions such that \(f_n \le f\)  for every \(n \in \mathbb {N}\)  and \(f_n\downarrow 0\)  \(\mu \) -a.e. [1]}. The set of all order-continuous elements in \(X\)  is denoted by \(X_a\) . The subspace \(X_a\)  is a closed subspace of \(X\)  and if \(|g| \le |f|\)  \(\mu \) -a.e. and \(f \in X_a\)  then \(g \in X_a\) . [1]}. The closure of the set of all simple functions of finite measure support is denoted by \(X_b\) . In general, we have \(X_a \subset X_b\)  [1]}.
